# Analysis of the Aerodynamics in the Heating Section of an Anode Baking Furnace Using Non-Linear Finite Element Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Geometry Definition

#### 2.2. Non-Isothermal Fuel Jets in Cross-Flow

#### 2.3. Mesh Generation

- Mesh obtained by cfMesh (Mesh 1) and COMSOL (Mesh 2)
- Mesh obtained by cfMesh (Mesh 1) with and without local refinement (Mesh 3)

#### 2.4. Governing Equations

#### 2.4.1. Flow Equation

#### 2.4.2. Standard k-$\u03f5$ Model

#### 2.4.3. Realizable k-$\u03f5$ Model

#### 2.4.4. Energy Equation

**q**is computed by Equation (22).

#### 2.5. Finite Element Discretization

#### 2.6. Pseudo-Time Stepping Solver with Non-Linear and Linear Equations

## 3. Results and Discussion

#### 3.1. Baseline Model with Mesh 1

#### 3.2. Non-Isothermal Effect on Baseline Model with Mesh 1

#### 3.3. Comparison of Mesh 1 and Mesh 2

#### 3.4. Comparison of Mesh 1 and Mesh 3

#### 3.5. Realizable k-$\u03f5$ Model

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RANS | Reynolds averaged Navier Stokes |

WSGGM | Weighted sum of gray gas model |

LES | Large eddy simulation |

DNS | Direct numerical simulation |

IGES | Initial graphics exchange specification |

STEP | Standard for the exchange of product Data |

MPI | Message passing interface |

CFL number | Courant Friedrichs Lewy number |

PID | Proportional-Integral-Derivative |

GMRES | generalized minimal residual method |

SAAMG | Smoothed aggregated algebraic multigrid |

GMG | Geometric multigrid |

MUMPS | Multifrontal Massively Parallel sparse |

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**Figure 1.**The schematic representation of the anode baking furnace. The left-most picture is the photograph from the actual furnace showing the exit of the flue channel. The right-most picture shows a photograph of the burner through which fuel is injected.

**Figure 2.**The schematic representation of different zones in the anode baking furnace based on the transfer of heat. The sections presented in the red box are in the heating zone. One section (section 7) is studied in this paper due to its importance with respect to the NOx generation.

**Figure 4.**The Cartesian mesh generated by the software cfMesh (Mesh 1) of the heating section of the anode baking furnace.

**Figure 5.**Mesh at the symmetry plane (Z = 0.27 m) in the flue wall prepared in (

**a**) cfMesh (Mesh 1) and (

**b**) COMSOL (Mesh 2) software respectively.

**Figure 6.**Mesh at the symmetry plane (Z = 0.27 m) in the heating section prepared in cfMesh (

**a**) with jet refinement (Mesh 1) and (

**b**) without jet refinement (Mesh 3).

**Figure 8.**The color plots of (

**a**) velocity (

**b**) turbulent viscosity ratio and (

**c**) temperature at the symmetry plane (Z = 0.27 m) of baseline model with Mesh 1.

**Figure 9.**The convergence plot for the three segregated groups of different physics for baseline model with Mesh 1. The convergence is plotted for the step 4 from Table 8.

**Figure 11.**The color plots of velocity magnitude at the symmetry plane (Z = 0.27 m) for (

**a**) uncoupled and (

**b**) coupled flow and energy equations for baseline model with Mesh 1.

**Figure 12.**The color plots of velocity magnitude at the symmetry plane (Z = 0.27 m) generated by (

**a**) cfMesh (Mesh 1) and (

**b**) COMSOL mesh (Mesh 2).

**Figure 13.**The color plots of temperature at the symmetry plane (Z = 0.27 m) generated by (

**a**) cfMesh (Mesh 1) and (

**b**) COMSOL mesh (Mesh 2).

**Figure 14.**The color plots of turbulent viscosity ratio at the symmetry plane (Z = 0.27 m) generated by (

**a**) cfMesh (Mesh 1) and (

**b**) COMSOL mesh (Mesh 2).

**Figure 15.**The color plots of velocity magnitude at the symmetry plane (Z = 0.27 m) generated by cfMesh software (

**a**) with jet refinement (Mesh 1) and (

**b**) without jet refinement (Mesh 3).

**Figure 16.**The color plots of turbulent viscosity ratio at the symmetry plane (Z = 0.27 m) generated with the mesh of cfMesh software (

**a**) with jet refinement (Mesh 1) and (

**b**) without jet refinement (Mesh 3).

**Figure 17.**The color plots of temperature at the symmetry plane (Z = 0.27 m) generated by cfMesh software (

**a**) with jet refinement (Mesh 1) and (

**b**) without jet refinement (Mesh 3).

Property | Symbol | Air Inlet | Fuel Inlets | Initial Condition |
---|---|---|---|---|

Velocity [m/s] | U${}_{ref}$ | 1.45 | 74 | 0 |

Temperature [K] | T | 1050 | 300 | 300 |

Re [-] | Re | 8500 | 13,000 | - |

Turbulent length scale [m] | L${}_{T}$ | 0.01 | 0.01 | - |

Turbulent intensity [%] | I${}_{T}$ | 0.05 | 0.05 | - |

Physical Property | Symbol | Values |
---|---|---|

Specific heat capacity [J/(kg×K)] | Cp | 1004.5 |

Ratio of specific heats [-] | $\gamma $ | 1.4 |

Prandtl number [-] | Pr | 0.73 |

Molecular viscosity [kg/(m×s)] | $\mu $ | 1.8×10${}^{-5}$ |

Label | Mesh Generation Tool | Characteristics |
---|---|---|

Mesh 1 | cfMesh | Refinement under burner |

Mesh 2 | COMSOL | Refinement under burner |

Mesh 3 | cfMesh | No Refinement under burner |

cfMesh | COMSOL Mesh | |
---|---|---|

Number of elements | 545,694 | 4,375,821 |

Average mesh quality | 0.861 | 0.666 |

Number of Cells | Average Mesh Quality | |
---|---|---|

Mesh with jet refinement | 545,694 | 0.86 |

Mesh without jet refinement | 470,048 | 0.89 |

Constant | Value |
---|---|

${C}_{\mu}$ | 0.09 |

${C}_{\u03f51}$ | 1.44 |

${C}_{\u03f52}$ | 1.92 |

${\sigma}_{k}$ | 1.00 |

${\sigma}_{\u03f5}$ | 1.30 |

Constant | Value |
---|---|

${C}_{\u03f52}$ | 1.90 |

${\sigma}_{k}$ | 1.00 |

${\sigma}_{\u03f5}$ | 1.20 |

${A}_{0}$ | 4.00 |

Simulation Step | CPU Time | Newton Iterations | GMRES Iterations (v,p) | GMRES Iterations (k,$\mathit{\u03f5}$) |
---|---|---|---|---|

step 1 | 1 h 35 min | 22 | 478 | 1674 |

step 2 | 2 h 11 min | 25 | 479 | 1812 |

step 3 | 1 h 6 min | 48 | 222 | 710 |

step 4 | 2 h 33 min | 121 | 1027 | 699 |

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**MDPI and ACS Style**

Nakate, P.; Lahaye, D.; Vuik, C.; Talice, M.
Analysis of the Aerodynamics in the Heating Section of an Anode Baking Furnace Using Non-Linear Finite Element Simulations. *Fluids* **2021**, *6*, 46.
https://doi.org/10.3390/fluids6010046

**AMA Style**

Nakate P, Lahaye D, Vuik C, Talice M.
Analysis of the Aerodynamics in the Heating Section of an Anode Baking Furnace Using Non-Linear Finite Element Simulations. *Fluids*. 2021; 6(1):46.
https://doi.org/10.3390/fluids6010046

**Chicago/Turabian Style**

Nakate, Prajakta, Domenico Lahaye, Cornelis Vuik, and Marco Talice.
2021. "Analysis of the Aerodynamics in the Heating Section of an Anode Baking Furnace Using Non-Linear Finite Element Simulations" *Fluids* 6, no. 1: 46.
https://doi.org/10.3390/fluids6010046